Find the Bogus Coin

Feb 17, 2018

RAY: Imagine that you have in front of you fifty coins. They all look exactly alike except one of them is a fake. Because it's a fake, it weighs a couple of grams more than a real coin. So, if you had a balance scale, and you knew which was the bogus coin, you would put it on one side of the scale, a good coin on the other side...

TOM: ...and it would be immediately obvious from this imbalance which was the phony coin, because it's heavier than a real coin.

RAY: Right. Knowing that, you have in front of you fifty coins -- one of which is bogus. The question is, what is the fewest number of weighings on a balance scale that you need to perform to determine which coin is bogus?

TOM: And-- Part B of the puzzler: Tell us how you got that number.

Answer:

RAY: The answer is 4 weighings. Now how did we get this number? At first blush, you would think, because of other puzzlers of this ilk, that you would divide the 50 coins in half and 50 is conveniently divided in half, right?

TOM: Yeah. So, you'd do 25 and 25. That's weighing number one.You find out that it's on the left side.

RAY: Then you do 12 and 12 with one leftover, and assume the worst case scenario, one of them's heavier.

TOM: Then six and six. That's three weighings. Three and three. That's four weighings. And you're done for. It takes five.

RAY: So, you had to come up with something a little more clever. And what you do is divide the coins into three piles. Two piles of 17 and one of 16.

And so, you take the two piles of 17 and you put those on the scale, and you keep the 16 pile aside, right?

Right away, you can see that you're going to eliminate not half the coins, but two thirds of the coins.So, let's assume that one of the 17 is the heavier one. You throw everything else away.

TOM: And you've only made one weighing. And you've narrowed it down to 17.

RAY: Now, you could divide the 17 in half, but better still, divide it thirds and you've got six and six and five. And that's the second weighing.

TOM: Then three and three. And one and one, and that's it.

RAY: And then, and bingo! And the key is, once you figure out the idea that you're going to divide it into three piles and not two, it jumps right out at you.